Stark conjectures

E839485

conjecture in algebraic number theory mathematical conjecture

The Stark conjectures are a set of deep conjectures in algebraic number theory that predict precise connections between special values of L-functions and the arithmetic of number fields, particularly units and class fields.

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All labels observed (2)

Label	Occurrences
Brumer–Stark conjecture	1
Stark conjectures canonical	1

Statements (45)

Predicate	Object
instanceOf	conjecture in algebraic number theory ⓘ mathematical conjecture ⓘ
concerns	leading terms of Artin L-functions at s = 0 ⓘ special values of L-functions at s = 0 ⓘ special values of L-functions at s = 1 ⓘ
field	algebraic number theory ⓘ
goal	describe arithmetic invariants via L-values ⓘ give explicit generators of abelian extensions ⓘ
hasPart	Stark conjecture for totally real fields NERFINISHED ⓘ Stark conjecture over Q ⓘ higher-rank Stark conjectures NERFINISHED ⓘ rank-one Stark conjecture NERFINISHED ⓘ
implies	existence of canonical units in certain extensions ⓘ explicit class field theory statements ⓘ
influenceOn	development of modern class field theory ⓘ research on special values of L-functions ⓘ
involves	Artin characters NERFINISHED ⓘ Dirichlet characters NERFINISHED ⓘ Galois representations ⓘ idele class groups ⓘ regulators of units ⓘ
mainTheme	arithmetic of number fields ⓘ class fields ⓘ special values of L-functions ⓘ units in number fields ⓘ
namedAfter	Harold Stark NERFINISHED ⓘ
predicts	connections between special L-values and units ⓘ construction of abelian extensions from L-values ⓘ existence of Stark units ⓘ relations between regulators and derivatives of L-functions ⓘ
proposedBy	Harold Stark NERFINISHED ⓘ
relatedTo	Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ Brumer–Stark conjecture NERFINISHED ⓘ Gross–Stark conjecture NERFINISHED ⓘ Leopoldt conjecture NERFINISHED ⓘ Rubin–Stark conjecture NERFINISHED ⓘ
relates	Artin L-functions NERFINISHED ⓘ L-functions ⓘ abelian extensions of number fields ⓘ class field theory ⓘ
status	open problem in mathematics ⓘ
timePeriod	20th century ⓘ
usedIn	Iwasawa theory NERFINISHED ⓘ construction of p-adic L-functions ⓘ explicit class field theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hilbert’s twelfth problem → relatedTo → Stark conjectures ⓘ

this entity surface form: Brumer–Stark conjecture